For questions about recurrence relations, convergence tests, and identifying sequences.

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39
votes
4answers
864 views

Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$

Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$ It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
30
votes
3answers
2k views

Why is this series of square root of twos equal $\pi$?

Wikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+... \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, like ...
12
votes
4answers
2k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
14
votes
3answers
881 views

How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an ...
11
votes
2answers
401 views

How to prove that $ \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | $ diverges?

It is stated as a problem in Spivak's Calculus and I can't wrap my head around it.
12
votes
3answers
900 views

Limit $\lim\limits_{n\rightarrow\infty}\left({2\sqrt n}-\sum\limits_{k=1}^n\frac{1}{\sqrt k}\right)$

How to find $\displaystyle\lim_{n\rightarrow\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac{1}{\sqrt k}\right)$ ? And generally does the limit of the integral of f(x) minus the sum of f(x) exist? How to ...
4
votes
4answers
6k views

How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate]

(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have ...
7
votes
4answers
678 views

Why does this expected value simplify as shown?

I was reading about the german tank problem and they say that in a sample of size $k$, from a population of integers from $1,\ldots,N$ the probability that the sample maximum equals $m$ is: ...
4
votes
6answers
516 views

Sequence sum question: $\sum_{n=0}^{\infty}nk^n$ [duplicate]

I am very confused about how to compute $$\sum_{n=0}^{\infty}nk^n.$$ Can anybody help me?
5
votes
2answers
407 views

Another Epsilon-N Limit Proof Question

How to prove the limit of the following sequence using epsilon-N argument. $$a_n=\frac{3n^2+2n+1}{2n^2+n}$$ I took the limit to be $\frac{3}{2}$ and proceeded with the argument, ...
86
votes
4answers
3k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
59
votes
4answers
3k views

How to sum this series for $\pi/2$ directly?

The sum of the series $$ \frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1} $$ can be derived by accelerating the Gregory Series $$ \frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2} ...
38
votes
3answers
1k views

Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?

Does the following series converge or diverge? I would like to see a demonstration. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$ I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
56
votes
4answers
6k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
20
votes
3answers
556 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$

Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
29
votes
4answers
938 views

A sum containing harmonic numbers $\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
17
votes
2answers
2k views

Limit of Nested Radical $\sqrt{1 + 2 \sqrt{1+3 \sqrt{1+ \cdots }}}$

How does one evaluate show that this limit: $$\lim_{n \to \infty}\sqrt{1 + 2 \sqrt{1+3 \sqrt{1+ \cdots \sqrt{1+(n-1) \sqrt{1+n}}}}}=3$$
21
votes
4answers
5k views

Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$

I recently came across this problem Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times }} = 3$ After ...
15
votes
3answers
391 views

$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$

I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ Since ...
19
votes
4answers
691 views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
17
votes
2answers
395 views

How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?

I'm considering the transfer-function $$ t(x) = \log(1 + \exp(x)) $$ and find the beginning of the power series (simply using Pari/GP) as $$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
12
votes
2answers
656 views

Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$

Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$ converge? I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
9
votes
5answers
2k views

Prove $\lim_{x \rightarrow 0} \frac {\sin(x)}{x} = 1$ with the epsilon-delta definition of limit.

It is well known that $$\lim_{x \rightarrow 0} \frac {\sin(x)}{x} = 1$$ I know several proofs of this: the geometric proof shows that $\cos(\theta)\leq\frac {\sin(\theta)}{\theta}\leq1$ and using ...
8
votes
5answers
527 views

Definite integral, quotient of logarithm and polynomial

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
7
votes
3answers
894 views

$1 + 1 + 1 +\cdots = -\frac{1}{2}$

The formal series $$ \sum_{n=1}^\infty 1 = 1+1+1+\dots=-\frac{1}{2} $$ comes from the analytical continuation of the Riemann zeta function $\zeta (s)$ at $s=0$ and it is used in String Theory. I am ...
13
votes
4answers
628 views

Do these series converge to logarithms?

It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: ...
11
votes
5answers
5k views

Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$

Is there any formula for this series? $$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
10
votes
3answers
14k views

Predict next number from a series

Which methods I can use to predict next number from a series of numbers ? I know the min & max possible number in advance.
15
votes
3answers
660 views

is a net stronger than a transfinite sequence for characterizing topology?

For metric spaces, knowledge of the convergence of sequences determines the topology completely. A set is closed in the metric topology if and only if it is closed under the limit of convergent ...
9
votes
2answers
738 views

Convergent series exercise from Little Rudin

Problem: Suppose that for every $n\in\mathbb{N}$, $a_n\in\mathbb{R}$ and $a_n\ge 0$. Given that $$\sum_0^\infty a_n$$ converges, show that $$\sum_1^\infty \frac{\sqrt{a_n}}{n} $$ ...
7
votes
2answers
4k views

If a series converges, then the sequence of terms converges to $0$.

Following the guidelines suggested in this meta discussion, I am going to post a proposed proof as an answer to the theorem below. I believe the proof works, but would appreciate any needed ...
8
votes
5answers
485 views

Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$

Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to \infty} s_n$. \begin{align} s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx \end{align} I have ...
14
votes
2answers
2k views

Do these series converge to the von Mangoldt function?

Jeffrey Shallit formulated this recurrence for me: $\displaystyle T(n,1)=1, k>1: T(n,k) = \sum\limits_{i=1}^{k-1} T(n-i,k-1)-\sum\limits_{i=1}^{k-1} T(n-i,k)$ which is the lower triangular array ...
5
votes
4answers
172 views

Proof of $\lim_{n \to \infty} {a_n}^{1/n} = \lim_{n \to \infty}(a_{n+1}/a_n)$

Why would $\lim_{n \to \infty} {a_n}^{1/n} = \lim_{n \to \infty}(a_{n+1}/a_n)$ be true where $(a_n)$ is a sequence in $\mathbb{R}$? Edit: Let all $a_n$ be positive.
12
votes
3answers
251 views

The set of functions which map convergent series to convergent series

Suppose $f$ is some real function with the above property, i.e. if $\sum\limits_{n = 0}^\infty {x_n}$ converges, then $\sum\limits_{n = 0}^\infty {f(x_n)}$ also converges. My question is: can ...
8
votes
2answers
66k views

Whats infinity divided by infinity?

This should be a simple question but i just want to make sure. I know from infinity/infinity is undefined. However if we have 2 equal infinities divided by each other it would be 1? And if we have ...
7
votes
2answers
208 views

Checking my understanding: $1 - 1 + 1 - 1 + 1 - … = \frac{1}{2}$

I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows: Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...
7
votes
5answers
1k views

Calculate the sum of the infinite series $\sum_{n=0}^{\infty} \frac{n}{4^n}$

A previous problem had us solving $\sum_{n=0}^{\infty} \frac{1}{4^n}$ which I calculated to be $\frac{4}{3}$ using a bit of mathematical manipulation. Wonderful. Thank you for all the prompt ...
6
votes
4answers
1k views

Showing inequality for harmonic series.

I want to show that $$\log N<\sum_{n=1}^{N}\frac{1}{n}<1+\log N.$$ But I don't know how to show this.
5
votes
2answers
320 views

Convergence of $\sum_{n=2}^\infty \frac{1}{n^\alpha \ln^\beta (n)} $

Study the convergence of the following series: $$\sum_{n=2}^\infty \frac{1}{n^\alpha \cdot\ln^\beta(n)} \text{ where }\alpha,\beta \geq 0 $$ Applying d'Alembert criterion I have that $$ ...
6
votes
2answers
2k views

Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$?

I can prove this sum has a constant upper bound like this: $$\sum_{k=1}^n \frac1{k ^ 2} \lt 1 + \sum_{k=2}^n \frac 1 {k (k - 1)} = 2 - \frac 1 n \lt 2$$ And computer calculation shows that sum ...
6
votes
4answers
2k views

Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...
2
votes
6answers
205 views

Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$

I got this question in my maths paper Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$ and find the sum if it exists. I managed to show that the series ...
1
vote
1answer
358 views

Taylor series for $\log(1+x)$ and its convergence

I know the taylor series of $\log(1+x)$. However I don't understand how to find the convergence for $x>1$ and divergence if $x<1$.
63
votes
2answers
2k views

How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$

Question: Show that $$A=\lim_{n\to \infty}\sqrt{1+\sqrt{\dfrac{1}{2}+\sqrt{\dfrac{1}{3}+\cdots+\sqrt{\dfrac{1}{n}}}}}$$ exists, and find the best estimate limit $A$. It is easy to show that ...
32
votes
6answers
919 views

A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$

Let $H_{n}$ be the nth harmonic number defined by $ H_{n} = \sum_{n=1}^{n} \frac{1}{k}$. I'm interested in knowing how to show that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}.$$ I tried ...
25
votes
3answers
590 views

$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2)$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
23
votes
4answers
726 views

Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$

How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ This is about as far as I got: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
7
votes
7answers
4k views

Series that converge to $\pi$ quickly

I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does ...
19
votes
4answers
2k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...